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Model Construction for Convex-Constrained Derivative-Free Optimization

Lindon Roberts

Abstract

We develop a new approximation theory for linear and quadratic interpolation models, suitable for use in convex-constrained derivative-free optimization (DFO). Most existing model-based DFO methods for constrained problems assume the ability to construct sufficiently accurate approximations via interpolation, but the standard notions of accuracy (designed for unconstrained problems) may not be achievable by only sampling feasible points, and so may not give practical algorithms. This work extends the theory of convex-constrained linear interpolation developed in [Hough & Roberts, SIAM J. Optim, 32:4 (2022), pp. 2552-2579] to the case of linear regression models and underdetermined quadratic interpolation models.

Model Construction for Convex-Constrained Derivative-Free Optimization

Abstract

We develop a new approximation theory for linear and quadratic interpolation models, suitable for use in convex-constrained derivative-free optimization (DFO). Most existing model-based DFO methods for constrained problems assume the ability to construct sufficiently accurate approximations via interpolation, but the standard notions of accuracy (designed for unconstrained problems) may not be achievable by only sampling feasible points, and so may not give practical algorithms. This work extends the theory of convex-constrained linear interpolation developed in [Hough & Roberts, SIAM J. Optim, 32:4 (2022), pp. 2552-2579] to the case of linear regression models and underdetermined quadratic interpolation models.
Paper Structure (14 sections, 14 theorems, 90 equations, 3 algorithms)

This paper contains 14 sections, 14 theorems, 90 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Existing work
  3. Contributions
  4. Organization
  5. Notation
  6. Background
  7. Model Construction & Accuracy
  8. Algorithm Specification
  9. Linear Regression Models
  10. Underdetermined Quadratic Interpolation Models
  11. Constructing $\bm{\Lambda}$-Poised Quadratic Models
  12. Updating the matrix $\bm{F}$
  13. Constructing $\bm{\Lambda}$-Poised Sets
  14. Conclusions and Future Work

Key Result

Theorem 2.6

If Assumptions ass_smoothness, ass_boundedhess and ass_cdec hold, then $\lim_{k\to\infty} \pi^f_k = 0$. Moreover, if $\epsilon\in(0,1]$ and $\epsilon_C \geq c_2 \epsilon$ for some constant $c_2>0$, then the number of iterations $k$ before Algorithm alg_cdfotr produces an iterate with $\pi^f_k < \eps

Theorems & Definitions (33)

  • Definition 2.4
  • Theorem 2.6: Theorem 3.10 & Corollary 3.15, LR_ConvexDFO2021
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Definition 4.1
  • Lemma 4.2
  • ...and 23 more